On Fourier Orthogonal Projections in the Rotation Algebra

Projections are constructed in the rotation algebra that areorthogonal to their Fourier transform and which are fixed underthe flip automorphism. Such projections are expected in a constructionof an inductive limit structure for the irrational rotationalgebra that is invariant under the Fourier transform (namely,as two circle algebras of the same dimension, which are swappedby the Fourier transform, plus a few points). The calculationis based on Rieffel's construction of the Schwartz space asan equivalence bimodule of rotation algebras as well as on thetheory of theta functions.     

Quantitation of trace levels of phenyltin compounds using HPTLC

Mixtures of tetraphenyltin, triphenyltin chloride, diphenyltin dichloride, and monophenyltin trichloride were resolved using high performance thin layer chromatography on silica gel with retention factor values of 0.80, 0.35, 0.20 and 0.01, respectively. Inorganic tin impurities were strongly adsorbed and did not migrate from the origin. Diphenyltin dichloride, monophenyltin trichloride, and inorganic tin components reacted with morin to produce fluorescent complexes. Postdevelopment exposure of the plate to ultraviolet light photodegraded the organic components which, after morin treatment, exhibited greater fluorescence than the organotins. This photolysis technique permitted the visualization of the otherwise non-fluorescent tetraphenyltin and weakly fluorescent triphenyltin spots.The components were quantitated using scanning densitometry. The working range varied from a maximum of 222 ng to a minimum of 1 ng, depending on the component and the excitation wavelength chosen. Thirty standards each containing five components were spotted, developed, derivatized, and scanned three times to produce 480 pieces of data within four hours. Calibration curves showed an instrumental error of 1.5% relative standard deviation units, and a spotter and intraplate variation of 9.0% relative standard deviation units. The inherent multiplicity of high performance thin layer chromatography allows for multiple sampling and analysis, thereby yielding significantly increased precision and high sample throughput. The chromatography and detection of butyltins and cyclohexyltins were also examined.     

Shear and extensional flow of polyacrylamide solutions

As part of an EEC Science Stimulation programme on extensional viscosity two major conferences were organised on the subject. The second of these was devoted to the results obtained on a standard fluid, M 1. The data obtained in shear flow was remarkably consistent from laboratory to laboratory. Extensional flow results presented quite a different picture. Using a series of nonequilibrium techniques such as the spinline rheometer, opposing jet, falling drop and converging flow, extensional viscosity results were obtained which differed by as much as two to three orders of magnitude. Nevertheless, it was apparent that consistancy did exist between similar techniques. It is in the context of this information that the measurements described below have been made.The shear and extensional flow properties of partially ionised polyacrylamide in solution at concentrations ranging from 5 ppm were measured. The method of solution preparation was found to have a profound effect on the behaviour of the solutions in shear flow. The influence of salt concentration and pH was investigated and is discussed in the context of molecular shape in solution.Extensional flow measurements, using the spinline rheometer, show that the solutions are strongly strain thickening even at concentrations as low as 5 ppm. These results are considered in the light of polymer entanglement and association in the strong flow field.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

A computational study of some rheological influences on the “splashing experiment”

In various attempts to relate the behaviour of highly-elastic liquids in complex flows to their rheometrical behaviour, obvious candidates for study have been the variation of shear viscosity with shear rate, the two normal stress differences N₁ and N₂, especially N₁, the extensional viscosity, and the dynamic moduli G′ and G″. In this paper, we shall confine attention to ‘constant-viscosity’ Boger fluids, and, accordingly, we shall limit attention to N₁, η_E, G′ and G″.We shall concentrate on the “splashing” problem (particularly that which arises when a liquid drop falls onto the free surface of the same liquid). Modern numerical techniques are employed to provide the theoretical predictions. We show that high η_E can certainly reduce the height of the so-called Worthington jet, thus confirming earlier suggestions, but other rheometrical influences (steady and transient) can also have a role to play and the overall picture may not be as clear as it was once envisaged. We argue that this is due in the main to the fact that splashing is a manifestly unsteady flow. To confirm this proposition, we obtain numerical simulations for the linear Jeffreys model.     

Iterated Point–Line Configurations Grow Doubly-Exponentially

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (Comput. Geom. 27:257–267, 2004) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemerédi–Trotter Theorem and an analysis of the “minimum degree” of the growing configuration.     

Nachweis und Bestimmung des Chroms

Ohne Zusammenfassung